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question is …....f(x+y)=f(x)+f(y) is a odd function prove..... i added image of solution of this question .please help me to understand solution …..i cant understand the solution which give in my book...........

question is …....f(x+y)=f(x)+f(y) is a odd function prove.....
i added image of  solution of this question .please help me to understand solution …..i cant understand the solution which give in my book...........

Question Image
Grade:12

5 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 years ago
Ans:
Hello Student,
Please find answer to your question below

f(x+y) = f(x)+f(y)
Solution which you attached is little bit complicated. So I am gonna do this by an easier method.
By definition,
f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}
f(x+h) = f(x)+f(h)
f'(x) = \lim_{h\rightarrow 0}\frac{f(x)+f(h)-f(x)}{h}
f'(x) = \lim_{h\rightarrow 0}\frac{f(h)}{h}
Since f(0) = 0, so it is zero by zero form. Apply L’hospital rule. We have,
f'(x) = f'(0)
\frac{df(x)}{dx} = f'(0)
\int df(x) = \int f'(0)dx
f(x) = f'(0)x + c
Put x = 0
f(0) = c
c = 0
f(x) = f'(0)x
So it is clearly odd function.
milind
23 Points
9 years ago
f`(x) = f`(0)\frac{df(x)}{dx} = f`(0)\int df(x) = \int f`(0)dx sir i can not understand this step please explain again fully
milind
23 Points
9 years ago
f`(x) = f`(0)\frac{df(x)}{dx} = f`(0)\int df(x) = \int f`(0)dx sir i can not understand this step please explain again fully
milind
23 Points
9 years ago
\int df(x) = \int f`(0)dx sir i can not understand this step please explain again fully \frac{df(x)}{dx} = f`(0)f`(x) = f`(0)
milind
23 Points
9 years ago
f`(x) = f`(0)….…..then\frac{df(x)}{dx} = f`(0)….…....then \int df(x) = \int f`(0)dx sir i can not understand this step please explain again fully

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